Homotopy coherent adjunctions and the formal theory of monads

Emily Riehl, Dominic Verity

    Research output: Contribution to journalArticleResearchpeer-review

    Abstract

    In this paper, we introduce a cofibrant simplicial category that we call the free homotopy coherent adjunction and characterise its n-arrows using a graphical calculus that we develop here. The hom-spaces are appropriately fibrant, indeed are nerves of categories, which indicates that all of the expected coherence equations in each dimension are present. To justify our terminology, we prove that any adjunction of quasi-categories extends to a homotopy coherent adjunction and furthermore that these extensions are homotopically unique in the sense that the relevant spaces of extensions are contractible Kan complexes. We extract several simplicial functors from the free homotopy coherent adjunction and show that quasi-categories are closed under weighted limits with these weights. These weighted limits are used to define the homotopy coherent monadic adjunction associated to a homotopy coherent monad. We show that each vertex in the quasi-category of algebras for a homotopy coherent monad is a codescent object of a canonical diagram of free algebras. To conclude, we prove the quasi-categorical monadicity theorem, describing conditions under which the canonical comparison functor from a homotopy coherent adjunction to the associated monadic adjunction is an equivalence of quasi-categories. Our proofs reveal that a mild variant of Beck's argument is "all in the weights"-much of it independent of the quasi-categorical context.

    LanguageEnglish
    Pages802-888
    Number of pages87
    JournalAdvances in Mathematics
    Volume286
    DOIs
    Publication statusPublished - 2 Jan 2016

    Fingerprint

    Adjunction
    Monads
    Homotopy
    Categorical
    Functor
    Free Algebras
    Nerve
    Justify
    Calculus
    Diagram
    Equivalence
    Closed
    Algebra
    Vertex of a graph
    Theorem

    Keywords

    • Adjunction
    • Homotopy coherence
    • Monad
    • Monadicity
    • Quasi-categories

    Cite this

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    Homotopy coherent adjunctions and the formal theory of monads. / Riehl, Emily; Verity, Dominic.

    In: Advances in Mathematics, Vol. 286, 02.01.2016, p. 802-888.

    Research output: Contribution to journalArticleResearchpeer-review

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